The value of integral_0^20pi (sin^4 x + cos^4 x)dx is equal to :

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q17 – JEE-MAIN 2026

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Home Maths IntegralsQuestion #206

The value of

\int_{0}^{20 π} (\sin^{4} x + \cos^{4} x) d x

is equal to :

Clue 1 of 3: Simplify the Integrand

Begin by simplifying the expression $\sin^{4} x + \cos^{4} x$ using trigonometric identities to make the integration more manageable.

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Step 1: Simplify the Integrand✦ Active

Rewrite the expression $\sin^{4} x + \cos^{4} x$ using algebraic and trigonometric identities.

\sin^{4} x + \cos^{4} x = (\sin^{2} x + \cos^{2} x)^{2} - 2 \sin^{2} x \cos^{2} x

= 1 - 2 (\sin x \cos x)^{2}

= 1 - 2 {(\frac{\sin (2 x)}{2})}^{2} = 1 - \frac{1}{2} \sin^{2} (2 x)

U s i n g \sin^{2} θ = \frac{1 - \cos (2 θ)}{2}, w e g e t :

= 1 - \frac{1}{2} (\frac{1 - \cos (4 x)}{2}) = 1 - \frac{1}{4} + \frac{1}{4} \cos (4 x) = \frac{3}{4} + \frac{1}{4} \cos (4 x)

Step 2: Integrate the Simplified Expression○ Expand

Substitute the simplified integrand back into the definite integral.

I = \int_{0}^{20 π} (\frac{3}{4} + \frac{1}{4} \cos (4 x)) d x

= \int_{0}^{20 π} \frac{3}{4} d x + \frac{1}{4} \int_{0}^{20 π} \cos (4 x) d x

Step 3: Evaluate the Definite Integral○ Expand

Calculate each part of the integral. The integral of $\cos (4 x)$ over $20 π$ (which is a multiple of its period $\frac{2 π}{4} = \frac{π}{2}$ ) is zero.

\int_{0}^{20 π} \frac{3}{4} d x = {[\frac{3}{4} x]}_{0}^{20 π} = \frac{3}{4} (20 π - 0) = 15 π

\int_{0}^{20 π} \cos (4 x) d x = 0

T h e r e f o r e, I = 15 π + \frac{1}{4} (0) = 15 π

💡 Teacher's Secret Hint

Remember that the integral of a cosine function over an interval that is an integer multiple of its period is zero.

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Maths Question 17 – JEE-MAIN 2026

Clue 1 of 3: Simplify the Integrand

🧠 Full Solution Path